2
− 1
,
.
If the variable a takes the value of the sapphire
lattice constant a
s
, Cu atoms of the interface will be
placed atop of the oxygen sites (ﬁg1c) and the considered
Cu structure in the metal/oxide system will be obtained.
Within the frame of the applied model, the interatomic
interactions are described by RGL potential. According-
ly, the potential energy per atom is a sum of the repulsive
energy E
rep
, and the binding energy E
b
:
∗
AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF MECHANICAL ENGINEERING AND ROBOTICS, DEPARTMENT OF THE STREGTH AND FATIGUE OF MATERIALS
AND STRUCTURES, 30-059 KRAKOW, 30 MICKIEWICZA AV., POLAND
512
Fig. 1. Periodic cell of Cu crystal in the metal/ oxide system (111)Cu || (0001)Al
2
O
3
: a) the close – packed plane of Cu with assumed
coordination system, b) periodic cell of Cu crystal assumed in ab initio calculations for the complex trigonal deformation path controlled by
the variable a. For a = a
0
= 4.428
˚
A fcc structure, for a = a
s
= 4.763
˚
A Cu structure in the metal oxide system c) the base of the periodic cell
of Cu crystal spanned on the oxygen atoms
E
rep
= A
¸
i
exp
¸
−p
¸
r
i
r
0
− 1

(1.2)
E
b
= −ξ

¸
i
exp
¸
−2q
¸
r
i
r
0
− 1

, (1.3)
where r
0
is the distance between nearest neighbors at ze-
ro temperature and r
i
is the distance of i atom from the
considered central one. The ﬁrst summand (1.2) is a pair
wise term, while the second one (1.3) is a many body
term. The binding energy is assumed as proportional to
the d band width. Thus the bond is formed by d elec-
trons without participation of s electrons, though s wave
functions localized in diﬀerent lattice points overlap each
other. Therefore, s electrons as well as d electrons are
present in common inter-node space ﬁg.2.
Fig. 2. Radial charge density distributions for 3d and 4s electrons coming from neighboring Cu atoms [7]
513
The considered model requires to determine four
parameters: A, ξ, p, q. In the present study, they are ob-
tained assuming that the model behaviour is correct in
the range of inﬁnitesimal strains but for arbitrary defor-
mation paths. Thus the proceeding of the strain process
ε is characterized by the function of energy density in
the form:
Φ

ε
i j

=
1
2
S
i jkl
ε
kl
ε
i j
, (1.4)
where S
i jkl
is elastic stiﬀness tensor.
In chapter 2, there is shown by means of group theory
and a direct physical analysis, how the crystal symmetry
controls the form of the stiﬀness tensor S and according-
ly uniquely determines the basis of the eigen-subspaces
of this tensor. The energy density stored in an arbitrary
strain process is a sum of the energy densities belonging
to three eigen-subspaces, as it was shown in the less
known papers by J. Rychlewski [8], [9]. Accordingly, if
the parameters A, ξ, p, q are chosen in such a way that
the model subjected to three strain states deﬁned by basis
vectors of three eigen-subspaces behaves correctly, the
employed assumption will be fulﬁlled.
To test the studied model the own program which
uses the code of the package for quantum-mechanical
calculations (CASTEP [10]) written. CASTEP is the imple-
mentation of Kohn-Sham method [11], which enables to
determine the total energy of the system formed by the
atoms of given electronic conﬁgurations. As a result of
application of the created program the dependence of
the elastic strain energy density on the parameter con-
trolling the process is obtained. The program enables to
carry out the simulations of the ﬁnite strain processes in
the case of the cubic crystals (chapter 3).The considered
atomic model is subjected to the process of tetragonal
deformation. As a result two functions Φ(ε) are obtained.
One of them base on RGL potential determined in the
present paper and the second one is used by Dimitriev
[12]. The functions of energy density Φ(ε) are compared
to the results of ab initio calculations obtained by means
of the program written by one of the authors [7].
2. Inﬂuence of the crystal symmetry on the form of
RGL potential
Individual atoms of given electron conﬁgurations,
forming an ideal crystal, arrange themselves in the most
energetically advantageous structure. The copper crystal
has the face centered cubic structure (fcc). This struc-
ture is characterized by the symmetry group O
h
(ﬁg.3),
in which the inversion center and the four-fold symmetry
axes parallel to y, z axes can be assumed as the gener-
ator elements [13]. In the three-dimensional space, the
following matrices represent them [13]:
Fig. 3. The symmetry elements of Cu elementary cell (the point group O
h
)
D(i) =
,
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
−1 0 0
0 −1 0
0 0 −1
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
, D

C
4y

=
,
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
0 0 1
0 1 0
−1 0 0
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
, D(C
4z
) =
,
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
0 −1 0
1 0 0
0 0 1
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
(2.1)
514
The elastic strain process is described by function
Φ (ε), where Φ – the elastic strain energy density. In the
range of small strains the harmonic approximation can
be used [14]. According to it:
Φ(ε) =
1
2
S
i jkl
ε
i j
ε
kl
(2.2)
S
i jkl
(i, j, k, l = 1, 2, 3) is the stiﬀness tensor. If the course
of the process is considered in the nine-dimensional
strain space, S will take the form of the second rank
tensor. According to Neumann’s principle a quantity de-
scribing a physical property of a crystal remains in-
variant with respect to the operations belonging to the
symmetry group characterizing the given crystal [15].
The representation of the symmetry operator U in the
nine-dimensional space can be created on the basis of
the representation of the symmetry operator U from the
three-dimensional space by means of Cartesian product:
D

(U) = D(U) ⊗ D(U) (2.3)
If D (U) is the unitary operator then D’ (U) will
preserve this property. Thus the stiﬀness tensor charac-
terizing the elastic behavior of the crystal has to fulﬁll
the following conditions:
D

(U) S D
T
(U) = S, U ∈ G (2.4)
where G is the point group of the considered crystal.
In the case of copper, the stiﬀness tensor is subjected
the restrictions associated with the symmetry elements: i,
C
4y
, C
4z
. Their representations D’ (U) can be formulated
on the basis of the relationships (2.1), (2.3). Satisfying
the conditions (2.4) the stiﬀness tensor takes the form:
S =

¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
(2.5)
So the tensor S has four independent components: S
1111
,
S
1122
, S
1212
, S
1221
. Symmetry of the strain tensor and the
relationship (2.2) impose on the stiﬀness tensor addi-
tional restrictions:
S
i jkl
= S
jikl
= S
i jlk
= S
kli j
(2.6)
Thus in the case of the copper crystal three independent
components: S
1111
, S
1122
, S
1212
are obtained.
As it is shown above, the form, which takes the
stiﬀness tensor with regard to the crystal symmetry, can
be obtained by means of representation theory. The oth-
er way is the analysis of the courses of the appropriate
strain processes. According to Neumann’s principle, the
element belonging to the symmetry group of the con-
sidered crystal associates the states, in which the strain
processes proceed analogically. Thus the energy density
stored in these states is identical. Due to this, the rela-
tions between the components of the stiﬀness tensor are
obtained. Let’s take into consideration the strain process
described by the tensor ε with one non-zero element ε
11
(ﬁg.4a). Under the inﬂuence of the symmetry axis C
4z
the state ε is transformed into ε’, in which ε

Ω, (2.21)
where r
( j)
i
is the distance of the i atom from the consid-
ered central one in the deformed crystal subjected to the
j deformation path, while Ω is the volume of the crystal
unit cell. Then the atomic model behaves correctly when
is subjected arbitrary inﬁnite small strains. Additionally
the equilibrium conditions is introduced:
dΦ
1
(ε)
dε

ε=0
= 0 (2.22)
The identiﬁed atomic RGL model behaves correctly if
the aforementioned conditions are satisﬁed.
The presented way of the parameters identiﬁcation with
use of the energy densities (2.21), related with three
eigen-spaces of the stiﬀness tensor S, gives the possi-
517
bility of parameter calculation with controlled relative
error. The advantage of the elastic eigen-states approach
is that it provides the high symmetry of deformed crys-
tal, what enables obtaining of the analytic formulae for
r
( j)
i
(ε). The symbolic and numerical calculations were
carried out with use of MATLAB. An example of cal-
culated parameter values for i = 5 are given as follows:
A=0.0855 eV, ξ=1.224 eV, p=10.960, q=2.278. They are
diﬀerent from the values assumed by Dimitriev et al. [2],
[3]. He used RGL potential identiﬁed by Hecquet et al.
[12], where the parameters are determined according to
assumption that: cohesive energy and lattice parameter
take the experimental values, while the elastic constants
are as near experimental ones as possible.
3. Veriﬁcation of RGL atomistic model by means of
ab initio simulations of strain processes
The functions of the energy density stored during
diﬀerent strain processes Φ (ε) obtained by means of ab
initio calculations are the base of veriﬁcation of creat-
ed atomic models. Accordingly, the own program has
been written, which calling the commercial program
CASTEP (Cambridge Serial Total Energy Package) [10]
performs the simulation of an arbitrary homogeneous
strain process for ideal metal crystals of the cubic sym-
metry [7]. The strain process is deﬁned by Green tensor,
which diagonal elements determine the relative elonga-
tions (stretches) of the edges of the elementary cell and
the oﬀ-diagonal elements correspond to the changes of
the angles between the edges (shears). As regards the
homogeneity, each elementary cell is subjected to an
identical deformation. Accordingly, the homogeneously
strained ideal crystal can be still represented by Bravais
lattice. On the basis of Green strain tensor, in the given
iteration step, the created program deﬁnes geometry of
the elementary cell with its symmetry and together with
the other parameters determined by user the program
introduces them as the input data into CASTEP code. The
additional parameters are: the cutoﬀ energy E
cut
, the
density of Monkhorst-Pack mesh and the type of the
exchange-correlation functional. In the performed simu-
lations the functional in a form proposed by J.P. Perdew
and Y.A. Wang (PW91) [16] obtained by the generalized
gradient approximation (GGA) is used. Because of the
way of operation of CASTEP software the own code is
the Linux shell script written in bash. As a result of the
program application one obtains the set of the total ener-
gies per the elementary cell volume E
c
corresponding to
the successive stages of the strain process. On the basis
of the calculated E
c
one gets the strain energy density
according to the formula [17]:
Φ(ε) =
E
c
(ε) − E
c

ε
eq

V
eq
(3.1)
where E
c
(ε
eq
) is the total energy of the ideal crystal in
the equilibrium state per the elementary cell of volume
V
eq
.
The developed method of the strain process simulation
was used to characterize the elastic behaviour of Cu crys-
tal in the range of small strains. In this aim, the ideal
Cu crystal has been subjected to the strain processes be-
longing to the particular eigen-subspaces of the stiﬀness
tensor. As a result the relations Φ (ε
i
), i=I, II, III were
obtained. Approximating them by parabolas, according
to the formulas (2.19), Kelvin moduli λ
I
, λ
II
, λ
III
were
determined. The calculation parameters: E
cut
, density of
Monkhorst-Pack mesh were selected in such a way to
get the total energies E
c
(ε) converged to less than 0.1
meV/atom. The comparison of the obtained results to the
experimental data [18] enabled to verify the developed
program.
In the case of the ﬁrst eigen-state the parameter ε
(2.17) is associated with the relative elongation of three
perpendicular edges of the elementary cell ∆a
z.I
a
z.eq
by the following formula:
ε =
√
3
2
,
¸
¸
¸
¸
¸
¸
¸
∆a
z.I
a
z.eq
+ 1

2
− 1
¸
¸
¸
¸
¸
¸
¸
(3.2)
During the strain process ∆a
z.I
= −0.04 0.03
˚
A with
the iteration step h = 0.001
˚
A. The symmetry of the el-
ementary cell doesn’t change and it is determined by
the symmetry elements shown at the ﬁg.3 The cutoﬀ
energy E
cut
=400eV and Monkhorst – Pack mesh den-
sity 13x13x13 were assumed for the calculations. The
total energy of the crystal per the elementary cell E
c
reaches the minimum at the equilibrium conﬁguration.
Thus using the relation E
c
(a
z.I
) obtained from the sim-
ulation, the equilibrium lattice constant a
z.eq
= 3.604
˚
A
was determined. This result diﬀers from experimental
one of 0.3%. The obtained equilibrium lattice constant
enables to present the simulation results in the form of
the relation Φ (ε
I
). Applying the least-squares method
Φ (ε
I
) has been approximated by a parabola, the dou-
bled coeﬃcient of which is Kelvin modulus λ
I
(ﬁg.6).
The simulation results reveal very good agreement with
the experiment [18]. The square of the correlation co-
eﬃcient R
2
=0.9981 and Kelvin modulus λ
I
=424.4 GPa
diﬀers from the experimental one of 0.4%.
518
Fig. 6. The elastic strain energy density Φ as a function of the parameter ε controlling the course of the process belonging to the ﬁrst
eigen-subspace of the stiﬀness tensor
Let’s take into account the ﬁrst of the considered
states belonging to the second eigen-subspace of the
stiﬀness tensor (2.15). The parameter (2.17) can be re-
placed by the relative elongation of one of the edges
lying in the base of the elementary cell ∆a
z.II
a
z.eq
ﬁg.3.
The two quantities are related as follows:
ε =
√
6
2
,
¸
¸
¸
¸
¸
¸
¸
∆a
z.II
a
z.eq
+ 1

2
− 1
¸
¸
¸
¸
¸
¸
¸
((3.3)
The course of the strain process is determined by ∆a
z.II
changing in the range −0.024 0.025
˚
A with the itera-
tion step h = 0.001
˚
A. According to Green strain tensor
ε
II.1
(2.17) the elongation of the edge of the elementary
cell base induces appropriate shortening of the perpen-
dicular edge. Thus, the elementary cell takes the form
(ﬁg.7), which symmetry is characterized by the group
D
4h
. In the considered range of changes of ∆a
z.II
it can be
assumed that the elongation induces double shortening.
The simulation of the described strain process has been
carried out at the following parameters: E
cut
=500eV
and Monhhorst-Pack mesh density 14x14x14. As a re-
sult one has obtained Kelvin modulus λ
II
=51.98GPa
(ﬁg.8). Comparing to the experimental value the error
amounts to 1.3%. The square of the correlation coeﬃ-
cient R
2
=0.9975.
Fig. 7. The symmetry elements of Cu elementary cell subjected to the strain process ε
I I.1
belonging to the second eigen-subspace (the point
group D
4h
)
519
Fig. 8. The elastic strain energy density Φ as a function of the parameter ε controlling the course of the process ε
I I.1
belonging to the second
eigen-subspace of the stiﬀness tensor
In the strain process determined by the ﬁrst
of three distinguished states belonging to the third
eigen-subspace of the stiﬀness tensor (2.16), there occurs
only change of the angle between two edges in the base
of elementary cell ∆ϕ
12
= 90
◦
− ϕ
12
(ﬁg.9) Thus, one
can relate the parameter ε (2.18) to cos(ϕ
12
) as follows:
ε =
√
2
2
sin ∆ϕ
12
(3.4)
The quantity is varied in the range −1.2
◦
, 1.2
◦
with
the iteration step h = 0.1
◦
. During the strain process, the
elementary cell takes the form of much lower symmetry
characterized by group C
2h
(ﬁg.9).
Fig. 9. The symmetry elements of Cu elementary cell subjected to the strain process ε
I I I.1
belonging to the third eigen-subspace (the point
group C
2h
)
Performing the simulation of the described process ana-
logically as in the case of the second eigen-subspace
one has obtained the results shown in ﬁg.10 and so
λ
III
=151.52 GPa. The error of determination of Kelvin
modulus amounts to 7.4% and R
2
=0.9999. Thus, de-
spite the low symmetry of the system in the current
conﬁguration and so lowered accuracy of the quantum –
mechanical calculations, the obtained results are highly
consistent with the experiment.
520
Fig. 10. The elastic strain energy density Φ as a function of the parameter ε controlling the course of the process ε
I I I.1
belonging to the
third eigen-subspace of the stiﬀness tensor
Let’s take into account another process belonging
to the third subspace. This process is deﬁned by Green
strain tensor ε
trig
being the linear combination of the
states presented in chapter 2:
ε
trig
=
ε
√
6

¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
0 1 1
1 0 1
1 1 0

¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
(3.5)
According to the tensor ε
trig
there occurs an identical
change of the angles between three perpendicular edges
of the elementary cell ∆ϕ. The parameter ε is related to
∆ϕ as follows:
ε =
√
6
2
sin ∆ϕ (3.6)
∆ϕ undergoes changes in the range −2
◦
, 2
◦
with the
iteration step h = 0.1
◦
.
Thus, during the process the cuboid is being replaced
by the rhombohedron. Accordingly, one obtains the el-
ementary cell of much higher symmetry (D
3d
- ﬁg.11)
then in the early considered process of change of one
angle.
Fig. 11. The symmetry elements of Cu elementary cell subjected to the strain process ε
trig
belonging to the third eigen-subspace (the point
group D
3d
)
521
Due to this fact, performing the simulation analogical as
previously, one can expect much better results. Accord-
ing to this one has obtained Kelvin modulus λ
III
=158.3
GPa (ﬁg.12) diﬀering from the experimental value only
of 3.5%.
Fig. 12. The elastic strain energy density Φ as a function of the parameter ε controlling the course of the process ε
trig
belonging to the third
eigen-subspace of the stiﬀness tensor
The results obtained in the case of the process be-
longing to the second subspace were used to verify the
considered RGL atomistic model. In this aim the model
was subjected to the tetragonal deformation. The calcu-
lations were carried out for two sets of parameters: A, ξ,
p, q. As a result the functions Φ(ε) depicted in ﬁg. 13
were obtained. Their comparison to the ab initio simu-
lation enables to draw the conclusion that the atomistic
model behaves correctly in the range of small strains.
However the potential RGL formulated in the present
work gives the better results. Unfortunately applying the
considered model in the range of large strains one is not
able to obtain the metastable phase (bcc) [19], which
should appear at ε
√
6/10. According to LCAO method
which allows to formulate the considered model the in-
teratomic bond should be created by d electrons as well
as s- electrons (ﬁg.2). The RGL potential neglects the
last ones. The considered model behaves correctly in the
range of small strains only with regard to suitable selec-
tion of parameters, what was shown in the present paper.
In the range of ﬁnite strains the interatomic bond should
base on s-d hybridization. Such a model is the better
approximation of interatomic interactions across copper
crystal and gives a chance for the correct description of
Cu – Cu interactions between interfacial copper mono-
layer and bulk copper in the case of Cu/Al
2
O
3
system.
The other solution is the application of RGL potential
identiﬁed in the present paper restricting oneself to small
strains. Dimitriev et al, in the case of bulk Cu, used pa-
rameters arranged by Hecquet et al. [12] and describing
the interaction between the interfacial Cu monolayer and
the rest of Cu crystal identiﬁed the parameters by means
of the rigid tensile test, where the strain reached to 3.6
[3]. They did not obtain satisﬁed results [4].
Fig. 13. Functions of the energy density Φ(ε) characterizing proceedings of the tetragonal strain processes. They are obtained by means of
the atomistic models and ab initio calculations, respectively
522
4. Conclusions
Using crystal symmetry the parameters of RGL po-
tential in the case of copper crystal were identiﬁed. One
showed that the formulated model behaved correctly in
the range of small strains. Accordingly it can describe
the interactions across bulk copper being a part of the
system: (111)Cu——(0001)Al
2
O
3
if the system is sub-
jected to small deformations. The data for the model
veriﬁcations were obtained by means of own program,
which using the commercial program CASTEP performs
the simulations of homogenous strain processes. In the
paper, there was shown that consideration of ﬁnite defor-
mations requires developing of a new model in which the
interatomic bond bases on s – d hybridization. This will
be the subject of further study. Similarly to the present
paper, in the new model, there will be taken into account
crystal symmetry by means of the representation of the
point group operating in the nine dimensional space of
strains. To formulate and test this model the developed
program for simulations of strain processes will be used.
Acknowledgements
This work was supported partly by the Polish State Com-
mittee for Scientiﬁc Research within the framework of the Project
PBZ KBN 102/T08/2003 and partly by the Project KomCerMet of
the EU Operational Program Innovative Economy, 2007-2013. The
CASTEP calculations were carried out in Academic Computer Cen-
ter CYFRONET of AGH University of Science and Technology in
Kraków, Poland.
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Received: 7 December 2008.

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